Theories of Elastic Continua and Crystal Lattice Theories

摘要

This paper is concerned with the relations between lattice and continuum theories of the elastic behavior of perfect crystals. In the case of a simple Bravais lattice, it is shown in what way the differential equations of motion of successively higher order 'strain gradient' theories of elasticity correspond to successively shorter-wave approximations to the difference equations of motion of the lattice and how the additional material constants, which appear in the gradient theories, are related to the force constants of the lattice. In the case of a lattice with a basis, a new theory of elasticity is given for the long wave approximation, as distinguished from the long wave, low frequency approximation. It is also shown how to obtain shorter-wave continuum approximations to the equations of motion of a lattice with a basis. Finally, for both types of lattice, it is shown how the difference equations of motion may be converted to differential equations which yield the same dispersion relations, as do the difference equations, for all wave lengths. (Author)